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We study the reformulation of integer linear programs by means of a mixedinteger linear program with fewer integer variables. Such reformulations can besolved efficiently with mixed integer linear programming techniques. We exhibitexamples that demonstrate how integer programs can be reformulated using farfewer integer variables. To this end, we introduce a generalization of totalunimodularity called the \emph{affine TU-dimension} of a matrix and studyrelated theory and algorithms for determining the affine TU-dimension of amatrix. We also present bounds on the number of integer variables needed torepresent certain integer hulls.

Mixed Integer Reformulations of Integer Programs and the Affine TU-dimension of a Matrix